In this paper, we derive several forms of the equity volatility as a function of the equity value, from the structural credit risk literature. We then propose a new jump to default model by taking the equity volatility to be of the form implied by the models of Leland (1994) and Leland & Toft (1996). This model involves a process we call the Dual-Jacobi process and which has explicit formulae for its moments. Gram-Charlier expansions are then applied to approximate bond and call prices. Our model generalizes Linetsky (2006) by incorporating a local volatility which is bounded below by a positive constant. This local volatility will decrease to a positive constant for increasing stock prices, making the stock process asymptotic to Geometric Brownian Motion (GBM). In this sence, our model is more realistic than the Constant Elasticity of Variance (CEV) models.

In general, contingent claims on assets which may default during the duration of the contract cannot be priced and hedged consistently. This is due to the fact that the possibility of a default event brings in an extra uncertain factor, and there are therefore too few assets to construct a hedge against all sources of uncertainty. In this paper we show that consistent pricing and hedging is still possible if we assume that (1) we can estimate the size of the loss in value (as a percentage) upon default and (2) default is the only non-systematic risk factor involved. Moreover, we show that the resulting formulas for prices and hedges do not depend on the intensity of the default process, but on a new riskfree intensity which is an explicit function of other parameters in the model, in contrast to most other models. We derive a simple tree method to implement the methodology that is proposed, and show how other pricing methods for claims on defaultable assets are linked to our method. 1

The paper introduces a Black&Cox-type structural model for credit default swaps. The existing literature on structural CDS pricing is extended by allowing a general functional form for the default barrier specified without reference to asset volatilities, dividend yields and interest rates. We develop a fast and robust algorithm to compute survival probabilities numerically. An empirical application suggests that the market-implied barrier is stable over time, with a possibly hump-shaped term structure. The implied barrier can be used for computing survival probabilities consistent with objective expectations of asset evolution, for pricing under counterparty risk, and for determining optimal corporate bond covenants.

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Abstract. We solve the pricing problem for perpetual American puts and calls on dividend-paying assets. The dependence of a dividend process on the underlying stochastic factor is fairly general: any non-decreasing function is admissible. The stochastic factor follows a Lévy process. This specification allows us to consider assets that pay no dividends at all when the level of the underlying factor (say, the assets of the firm) is too low, and assets that pay dividends at a fixed rate when the underlying stochastic process remains in some range. Certain dividend processes exhibiting mean-reverting features can be modelled as appropriate increasing functions of Lévy processes. The pay-offs of both the American put and call options can be represented as the expected present value (EPV) of a certain stream of dividends: g(Xt) = δ(Xt) −qK and g(Xt) = qK − δ(Xt), respectively, and we show that the option must be exercised the first time the EPV of the stream g(X t), where X t = inf0≤s≤t Xs is the infimum process starting from the current level X0, becomes positive. Thus, the exercise threshold depends only on the record setting bad news. The results can be applied to the theory of real options as well; as one of possible applications, we consider the problem of incremental capital expansion. 1.

We give a review of the state of the art with regard to the theory of scale functions for spectrally negative Lévy processes. From this we introduce a general method for generating new families of scale functions. Using this method we introduce a new family of scale functions belonging to the Gaussian Tempered Stable Convolution (GTSC) class. We give particular emphasis to special cases as well as cross-referencing their analytical behaviour against known general considerations.

Motivated by classical considerations from risk theory, we investigate boundary crossing problems for refracted Lévy processes. The latter is a Lévy process whose dynamics change by subtracting off a fixed linear drift (of suitable size) whenever the aggregate process is above a pre-specified level. More formally, whenever it exists, a refracted Lévy process is described by the unique strong solution to the stochastic differential equation dUt = −δ1{Ut>b}dt + dXt where X = {Xt: t ≥ 0} is a Lévy process with law P and b, δ ∈ R such that the resulting process U may visit the half line (b, ∞) with positive probability. We consider in particular the case that X is spectrally negative and establish a suite of identities for the case of one and two sided exit problems. All identities can be written in terms of the q-scale function of the driving Lévy process and its perturbed version describing motion above the level b. We remark on a number of applications of the obtained identities to (controlled) insurance risk processes.

We give a review of the state of the art with regard to the theory of scale functions for spectrally negative Lévy processes. From this we introduce a general method for generating new families of scale functions. Using this method we introduce a new family of scale functions belonging to the Gaussian Tempered Stable Convolution (GTSC) class. We give particular emphasis to special cases as well as cross-referencing their analytical behaviour against known general considerations.

A thesis submitted for the transfer of status from